86 1. ALGEBRAIC THEORY OF COMPLEX MULTIPLICATION
1.7.5.1. Theorem (Yu). Let K be a field with positive characteristic, and A a CM
abelian variety over K with CM structure provided by a CM field L
End0(A).
If
OL End(A) then there is a finite extension K /K such that AK equipped with
its OL-action descends to a finite field contained in K .
This result is [133, Thm. 1.3]; it will not be used in what follows. (Note that
it suffices just to descend the abelian variety AK to a finite subfield of K , as then
a further finite extension on K will enable us to descend the abelian variety along
with its OL-action, by Lemma 1.2.1.2.)
1.8. CM lifting questions
1.8.1. Basic definitions and examples. Let κ be a field of characteristic p 0,
and consider an abelian variety A0 over κ. By Corollary 1.6.2.5, if κ is finite and
A0 is isotypic then we may endow it with a structure of CM abelian variety having
complex multiplication by a CM field. Inspired in part by Theorem 1.6.5.1, we
wish to pose several questions related to the problem of lifting A0 to characteristic
0 in the presence of CM structures. First we make a general definition unrelated
to complex multiplication.
1.8.1.1. Definition. A lifting of A0 to characteristic 0 is a triple (R, A, φ) con-
sisting of a domain R of characteristic 0, an abelian scheme A over R, a surjective
map R κ, and an isomorphism φ : A0 of abelian varieties over κ.
We may replace R with its localization at the maximal ideal ker(R κ) so
that R is local with residue field κ. For K := Frac(R), if AK admits sufficiently
many complex multiplications then we say A is a CM lift of A0 to characteristic 0.
The injective map End(A) End(AK ) has torsion-free cokernel:
1.8.2. Lemma. For abelian schemes A, B over an integral scheme S with generic
point η, the injective map Hom(A, B) Hom(Aη,Bη) has torsion-free cokernel.
Proof. Consider f : such that n · f extends to an S-group map
h : A B for a non-zero integer n. The restriction h : A[n] B[n] between finite
flat S-groups vanishes because such vanishing holds on the generic fiber over the
integral S. Since [n] : A A is an fppf covering with kernel A[n], it follows that h
factors through this map over S, which is to say h = n · f for some S-group map
f : A B. Hence, the map f Hom(Aη,Bη) is killed by n, so = f.
The injective map in Lemma 1.8.2 can fail to be surjective:
1.8.3. Example. Let p be a prime with p 3 (mod 4), so p is prime in Z[i] (with
i2
= −1). Let R = Z(p) +pZ(p)[i], so [Z(p)[i] : R] = p and Frac(R) = Q(i). Let E be
the elliptic curve y2 = x3 x over R, so the generic fiber EQ(i) has endomorphism
ring Z[i] via the action [i](x, y) = (−x, −iy). Because [i]∗(dx/y) = i · dx/y, Z[i]
acts on Lie(EQ(i)) through scaling via the canonical inclusion Z[i] Q(i).
We claim that End(E) = Z (so
End0(E)
:= Q⊗Z End(E) = Q, even though the
generic fiber EQ(i) has endomorphism algebra Q(i)). Indeed, if not then End(E) is
an order in Z[i] = End(EQ(i)), so End(E) = Z[i] by Lemma 1.8.2. In particular,
the action by i on EQ(i) would extend to an action on E, so the resulting multiplier
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